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BGK model for rarefied gas in a bounded domain

Hongxu Chen, Christian Klingenberg, Marlies Pirner

Abstract

We study the Bathnagar-Gross-Krook (BGK) equation in a smooth bounded domain featuring a diffusive reflection boundary condition with general collision frequency. We prove that the BGK equation admits a unique global solution with an exponential convergence rate if the initial condition is a small perturbation around the global Maxwellian in the $L^\infty$ space. For the proof, we utilize the dissipative nature from the linearized BGK operator and establish an $L^2$ coercive estimate. Next, we derive the a priori estimate by obtaining an $L^\infty$ bound on the nonlinear operator; this requires a delicate analysis to manage its intrinsic nonlinear structure. Finally, we establish the $L^\infty$ stability estimate and introduce sequential arguments for the nonlinear BGK operator, thereby concluding both well-posedness and positivity.

BGK model for rarefied gas in a bounded domain

Abstract

We study the Bathnagar-Gross-Krook (BGK) equation in a smooth bounded domain featuring a diffusive reflection boundary condition with general collision frequency. We prove that the BGK equation admits a unique global solution with an exponential convergence rate if the initial condition is a small perturbation around the global Maxwellian in the space. For the proof, we utilize the dissipative nature from the linearized BGK operator and establish an coercive estimate. Next, we derive the a priori estimate by obtaining an bound on the nonlinear operator; this requires a delicate analysis to manage its intrinsic nonlinear structure. Finally, we establish the stability estimate and introduce sequential arguments for the nonlinear BGK operator, thereby concluding both well-posedness and positivity.

Paper Structure

This paper contains 3 sections, 3 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Derivation of $\mathcal{L}$ and $\Gamma$

Key Result

Theorem 1

Assume $\Omega$ is bounded and smooth. There exists a constant $0<\delta\ll 1$ such that if the initial condition $F_0(x,v)=\mu+\sqrt{\mu}f_0(x,v)\geq 0$ satisfies $\int_{\Omega}\int_{\mathbb{R}^3}\sqrt{\mu}f_0(x,v)\,\mathrm{d} v\mathrm{d} x=0$ and then there exists a unique solution $F(t,x,v)=\mu+\sqrt{\mu}f(t,x,v)\geq 0$ to the problem f_eqn such that $\int_{\Omega}\int_{\mathbb{R}^3}\sqrt{\mu}

Theorems & Definitions (6)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Lemma 1: Yun
  • Lemma 2
  • proof